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Perfectoid fields, deeply ramified fields and their relatives - - PowerPoint PPT Presentation

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Perfectoid fields, deeply ramified fields and their relatives Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) ALaNT 5 B edlewo, June 2018 Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields Positive and

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Perfectoid fields, deeply ramified fields and their relatives

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) ALaNT 5 B¸ edlewo, June 2018

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field, and by O its valuation ring.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field, and by O its valuation ring. We are interested in valued fields (K, v) in the following two cases:

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field, and by O its valuation ring. We are interested in valued fields (K, v) in the following two cases: positive characteristic: char K = char Kv = p > 0,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 7

Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field, and by O its valuation ring. We are interested in valued fields (K, v) in the following two cases: positive characteristic: char K = char Kv = p > 0, mixed characteristic: char K = 0, char Kv = p > 0.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Positive and mixed characteristic

Given a valued field (K, v), we denote by vK its value group, by Kv its residue field, and by O its valuation ring. We are interested in valued fields (K, v) in the following two cases: positive characteristic: char K = char Kv = p > 0, mixed characteristic: char K = 0, char Kv = p > 0. In the following, p will always be the characteristic of the residue field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields, via the tilting construction.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields, via the tilting construction. A valued field (K, v) with char Kv = p > 0 is called perfectoid if it is complete,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields, via the tilting construction. A valued field (K, v) with char Kv = p > 0 is called perfectoid if it is complete, with p-divisible value group of rank 1 (i.e., vK is a subgroup of R),

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields, via the tilting construction. A valued field (K, v) with char Kv = p > 0 is called perfectoid if it is complete, with p-divisible value group of rank 1 (i.e., vK is a subgroup of R), and the homomorphism OK/pOK ∋ x → xp ∈ OK/pOK (1) is surjective.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Perfectoid fields

A modern tool for transferring information between the mixed and the positive characteristic case are the perfectoid fields, via the tilting construction. A valued field (K, v) with char Kv = p > 0 is called perfectoid if it is complete, with p-divisible value group of rank 1 (i.e., vK is a subgroup of R), and the homomorphism OK/pOK ∋ x → xp ∈ OK/pOK (1) is surjective. In the positive characteristic case, the latter condition is equivalent to K being perfect.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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What is interesting about perfectoid fields?

  • Perfectoid fields that correspond to each other through the

tilting construction have the same absolute Galois groups.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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What is interesting about perfectoid fields?

  • Perfectoid fields that correspond to each other through the

tilting construction have the same absolute Galois groups.

  • Perfectoid fields provide the basis for Scholze’s perfectoid

spaces.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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What is interesting about perfectoid fields?

  • Perfectoid fields that correspond to each other through the

tilting construction have the same absolute Galois groups.

  • Perfectoid fields provide the basis for Scholze’s perfectoid

spaces.

  • Perfectoid fields caught our interest in connection with our

work on the defect of valued field extensions.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] ,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] , where ν ≥ 0 is an integer.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 24

Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] , where ν ≥ 0 is an integer. (If the characteristic of the residue fields is 0, the formula remains true if we set p = 1.)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] , where ν ≥ 0 is an integer. (If the characteristic of the residue fields is 0, the formula remains true if we set p = 1.) The factor d(L|K, v) = pν is called the defect of the extension (L|K, v).

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] , where ν ≥ 0 is an integer. (If the characteristic of the residue fields is 0, the formula remains true if we set p = 1.) The factor d(L|K, v) = pν is called the defect of the extension (L|K, v). If pν > 1, then (L|K, v) is called a defect extension.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Defect

By (L|K, v) we denote a field extension L|K where v is a valuation on L and K is endowed with the restriction of v. If (L|K, v) is a finite extension of valued fields and the valuation v of K admits a unique extension to the field L, then by the Lemma of Ostrowski, [L : K] = pν · (vL : vK)[Lv : Kv] , where ν ≥ 0 is an integer. (If the characteristic of the residue fields is 0, the formula remains true if we set p = 1.) The factor d(L|K, v) = pν is called the defect of the extension (L|K, v). If pν > 1, then (L|K, v) is called a defect extension. If pν = 1, then we call (L|K, v) a defectless extension.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case, we call a separable defect extension of degree p dependent

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case, we call a separable defect extension of degree p dependent if it can be obtained from a purely inseparable defect extension by a simple transformation;

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case, we call a separable defect extension of degree p dependent if it can be obtained from a purely inseparable defect extension by a simple transformation;

  • therwise, we call it independent.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case, we call a separable defect extension of degree p dependent if it can be obtained from a purely inseparable defect extension by a simple transformation;

  • therwise, we call it independent.

Recently, we have been able to generalize this definition to the mixed characteristic case.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Types of separable defect extensions of degree p

In the positive characteristic case, we call a separable defect extension of degree p dependent if it can be obtained from a purely inseparable defect extension by a simple transformation;

  • therwise, we call it independent.

Recently, we have been able to generalize this definition to the mixed characteristic case. Anna Blaszczok will report on this in more detail in her talk.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 36

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 37

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 38

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),
  • decidability of Laurent Series Fields over finite fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 39

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),
  • decidability of Laurent Series Fields over finite fields.

In the positive characteristic case, a perfect valued field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 40

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),
  • decidability of Laurent Series Fields over finite fields.

In the positive characteristic case, a perfect valued field (such as Fp((t))1/p∞)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 41

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),
  • decidability of Laurent Series Fields over finite fields.

In the positive characteristic case, a perfect valued field (such as Fp((t))1/p∞) has no dependent defect extensions.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 42

The importance of this classification

There are several results (M. Temkin, S. ElHitti, L. Ghezzi and

  • S. D. Cutkosky, FVK) which indicate that the dependent defect

is more harmful than the independent defect for the solution of

  • pen problems such as
  • local uniformization (local form of resolution of singularities),
  • decidability of Laurent Series Fields over finite fields.

In the positive characteristic case, a perfect valued field (such as Fp((t))1/p∞) has no dependent defect extensions. What about perfectoid fields in mixed characteristic?

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 43

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our purposes.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 44

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 45

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete, and the property of

having rank 1

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 46

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete, and the property of

having rank 1 are both not first order axiomatizable.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 47

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete, and the property of

having rank 1 are both not first order axiomatizable. It is better to work with deeply ramified fields

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-48
SLIDE 48

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete, and the property of

having rank 1 are both not first order axiomatizable. It is better to work with deeply ramified fields in the sense of the book “Almost ring theory”

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-49
SLIDE 49

Shortcomings of perfectoid fields

It turns out that perfectoid fields are a bit too special for our

  • purposes. The property of being complete, and the property of

having rank 1 are both not first order axiomatizable. It is better to work with deeply ramified fields in the sense of the book “Almost ring theory” by O. Gabber and L. Ramero.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 50

Deeply ramified fields

Following Gabber and Ramero,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 51

Deeply ramified fields

Following Gabber and Ramero, a valued field (K, v) is deeply ramified if

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 52

Deeply ramified fields

Following Gabber and Ramero, a valued field (K, v) is deeply ramified if ΩOKsep|OK = 0 , (2)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 53

Deeply ramified fields

Following Gabber and Ramero, a valued field (K, v) is deeply ramified if ΩOKsep|OK = 0 , (2) where OK is the valuation ring of K,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 54

Deeply ramified fields

Following Gabber and Ramero, a valued field (K, v) is deeply ramified if ΩOKsep|OK = 0 , (2) where OK is the valuation ring of K, OKsep is the valuation ring

  • f the separable-algebraic closure of K,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 55

Deeply ramified fields

Following Gabber and Ramero, a valued field (K, v) is deeply ramified if ΩOKsep|OK = 0 , (2) where OK is the valuation ring of K, OKsep is the valuation ring

  • f the separable-algebraic closure of K, and ΩB|A denotes the

module of relative differentials when A is a ring and B is an A-algebra.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 57

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 58

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified if and only if its completion is perfect.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 59

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified if and only if its completion is perfect. In the mixed characteristic case, (K, v) is deeply ramified

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 60

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified if and only if its completion is perfect. In the mixed characteristic case, (K, v) is deeply ramified if and only if the value vp is not the smallest positive value in vK

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 61

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified if and only if its completion is perfect. In the mixed characteristic case, (K, v) is deeply ramified if and only if the value vp is not the smallest positive value in vK and OK/pOK ∋ x → xp ∈ OK/pOK is surjective

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 62

Characterization of rank 1 deeply ramified fields

Theorem (Gabber & Ramero) Take a valued field (K, v) of rank 1. In the positive characteristic case, (K, v) is deeply ramified if and only if its completion is perfect. In the mixed characteristic case, (K, v) is deeply ramified if and only if the value vp is not the smallest positive value in vK and OK/pOK ∋ x → xp ∈ OK/pOK is surjective (“the Frobenius on OK is surjective modulo p”).

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 63

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-64
SLIDE 64

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-65
SLIDE 65

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups that makes no sense for our purposes

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-66
SLIDE 66

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups that makes no sense for our purposes (namely, no archimedean component is discrete).

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-67
SLIDE 67

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups that makes no sense for our purposes (namely, no archimedean component is discrete). Therefore, we have introduced the larger class of generalized deeply ramified fields (gdr fields)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-68
SLIDE 68

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups that makes no sense for our purposes (namely, no archimedean component is discrete). Therefore, we have introduced the larger class of generalized deeply ramified fields (gdr fields) by taking the characterizations of the theorem of Gabber & Ramero

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-69
SLIDE 69

Shortcomings of deeply ramified fields

Gabber & Ramero’s characterization of deeply ramified fields

  • f higher rank is more complicated and involves an additional

property of the value groups that makes no sense for our purposes (namely, no archimedean component is discrete). Therefore, we have introduced the larger class of generalized deeply ramified fields (gdr fields) by taking the characterizations of the theorem of Gabber & Ramero as a definition in arbitrary rank.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 70

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 71

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 72

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise. Theorem Take a valued field (K, v) with char Kv = p > 0.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 73

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise. Theorem Take a valued field (K, v) with char Kv = p > 0. Then (K, v) is a gdr field if and only if

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 74

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise. Theorem Take a valued field (K, v) with char Kv = p > 0. Then (K, v) is a gdr field if and only if (vK)vp is p-divisible,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 75

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise. Theorem Take a valued field (K, v) with char Kv = p > 0. Then (K, v) is a gdr field if and only if (vK)vp is p-divisible, Kv is perfect,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 76

Gdr fields and defect

We denote by (vK)vp the smallest convex subgroup of vK that contains vp if char K = 0, and set (vK)vp = vK otherwise. Theorem Take a valued field (K, v) with char Kv = p > 0. Then (K, v) is a gdr field if and only if (vK)vp is p-divisible, Kv is perfect, and every separable defect extension of degree p is independent.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 77

Tame extensions

A finite extension (L|K, v) is called tame

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 78

Tame extensions

A finite extension (L|K, v) is called tame if it satisfies the following conditions:

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 79

Tame extensions

A finite extension (L|K, v) is called tame if it satisfies the following conditions: (T1) the ramification index (vL : vK) is not divisible by char Kv,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 80

Tame extensions

A finite extension (L|K, v) is called tame if it satisfies the following conditions: (T1) the ramification index (vL : vK) is not divisible by char Kv, (T2) the residue field extension Lv|Kv is separable,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 81

Tame extensions

A finite extension (L|K, v) is called tame if it satisfies the following conditions: (T1) the ramification index (vL : vK) is not divisible by char Kv, (T2) the residue field extension Lv|Kv is separable, (T3) [L : K] = (vL : vK)[Lv : Kv].

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 82

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 83

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 84

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 85

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect, have p-divisible value group (if the residue field has characteristic p > 0)

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 86

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect, have p-divisible value group (if the residue field has characteristic p > 0) and perfect residue field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 87

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect, have p-divisible value group (if the residue field has characteristic p > 0) and perfect residue field. It follows from condition (T3) that tame fields do not admit any defect extensions.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 88

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect, have p-divisible value group (if the residue field has characteristic p > 0) and perfect residue field. It follows from condition (T3) that tame fields do not admit any defect extensions. A henselian valued field (K, v) is called a separably tame field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 89

Tame and separably tame fields

A henselian valued field (K, v) is called a tame field if every finite extension is tame. All tame fields are perfect, have p-divisible value group (if the residue field has characteristic p > 0) and perfect residue field. It follows from condition (T3) that tame fields do not admit any defect extensions. A henselian valued field (K, v) is called a separably tame field if every finite separable extension is tame.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 90

Properties of tame fields

Tame fields have many good properties. For example:

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 91

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 92

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv. If char K = char Kv, then also relative completeness holds.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 93

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv. If char K = char Kv, then also relative completeness holds. Similar results hold for separably tame fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 94

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv. If char K = char Kv, then also relative completeness holds. Similar results hold for separably tame fields. Valued function fields over separably tame fields have a relatively good structure theory.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 95

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv. If char K = char Kv, then also relative completeness holds. Similar results hold for separably tame fields. Valued function fields over separably tame fields have a relatively good structure theory. This is used to prove the above theorem,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 96

Properties of tame fields

Tame fields have many good properties. For example: Theorem (K) Tame fields (K, v) satisfy model completeness and decidability relative to the elementary theories of their value groups vK and their residue fields Kv. If char K = char Kv, then also relative completeness holds. Similar results hold for separably tame fields. Valued function fields over separably tame fields have a relatively good structure theory. This is used to prove the above theorem, and it also has been applied to the problem of local uniformization.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 97

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 98

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 99

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3).

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 100

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3). However, as a deeply ramified field,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 101

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3). However, as a deeply ramified field, it does not admit extensions with dependent defect.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 102

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3). However, as a deeply ramified field, it does not admit extensions with dependent defect. The extension generated by a root of the Artin-Schreier polynomial Xp − X − 1/t has independent defect.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 103

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3). However, as a deeply ramified field, it does not admit extensions with dependent defect. The extension generated by a root of the Artin-Schreier polynomial Xp − X − 1/t has independent defect. The important question arises whether results on tame fields

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 104

Beyond tame fields

Note that Fp((t))1/p∞ is henselian and perfect, hence deeply ramified, but admits finite extensions that do not satisfy (T3). However, as a deeply ramified field, it does not admit extensions with dependent defect. The extension generated by a root of the Artin-Schreier polynomial Xp − X − 1/t has independent defect. The important question arises whether results on tame fields can be generalized to deeply ramified or even gdr fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 105

Semitame fields

To answer this question, we may want to stay,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 106

Semitame fields

To answer this question, we may want to stay, at least initially,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 107

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 108

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields, meaning that we only relax

  • ur conditions by allowing independent defect estensions.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 109

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields, meaning that we only relax

  • ur conditions by allowing independent defect estensions. We

call (K, v) a semitame field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 110

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields, meaning that we only relax

  • ur conditions by allowing independent defect estensions. We

call (K, v) a semitame field if it is a gdr field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-111
SLIDE 111

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields, meaning that we only relax

  • ur conditions by allowing independent defect estensions. We

call (K, v) a semitame field if it is a gdr field and its value group is p-divisible (if char Kv = p > 0).

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-112
SLIDE 112

Semitame fields

To answer this question, we may want to stay, at least initially, relatively close to the tame fields, meaning that we only relax

  • ur conditions by allowing independent defect estensions. We

call (K, v) a semitame field if it is a gdr field and its value group is p-divisible (if char Kv = p > 0). Semitame fields are our best bet when it comes to generalizing the results we have proved earlier for tame fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 113

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0,

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 114

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0, then the following logical relations between its properties hold:

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 115

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0, then the following logical relations between its properties hold: tame field ⇒ separably tame field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 116

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0, then the following logical relations between its properties hold: tame field ⇒ separably tame field ⇒ semitame field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 117

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0, then the following logical relations between its properties hold: tame field ⇒ separably tame field ⇒ semitame field ⇒ deeply ramified field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 118

A hierarchy of fields

Theorem 1) If (K, v) is a nontrivially valued field with char Kv = p > 0, then the following logical relations between its properties hold: tame field ⇒ separably tame field ⇒ semitame field ⇒ deeply ramified field ⇒ gdr field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 119

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce:

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 120

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 121

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 122

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 123

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 124

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult. In our work we have shown that to prove the above extension theorem

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 125

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult. In our work we have shown that to prove the above extension theorem it suffices to prove that every defect extension of degree p of a gdr field is again a gdr field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 126

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult. In our work we have shown that to prove the above extension theorem it suffices to prove that every defect extension of degree p of a gdr field is again a gdr field. We hope that this can be done directly

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

slide-127
SLIDE 127

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult. In our work we have shown that to prove the above extension theorem it suffices to prove that every defect extension of degree p of a gdr field is again a gdr field. We hope that this can be done directly by studying the behaviour of valuation rings

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 128

Algebraic extensions of deeply ramified fields

From Gabber & Ramero’s definition of deeply ramified fields the following result is easy to deduce: Theorem (Gabber & Ramero) Algebraic extensions of deeply ramified fields are again deeply ramified fields. The same follows for gdr fields via the characterization theorem of Gabber & Ramero. However, the proof of that theorem is difficult. In our work we have shown that to prove the above extension theorem it suffices to prove that every defect extension of degree p of a gdr field is again a gdr field. We hope that this can be done directly by studying the behaviour of valuation rings under such defect extensions.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 129

Independent defect fields

We call a valued field an independent defect field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 130

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 131

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 132

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect fields.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 133

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect

  • fields. For instance, is every algebraic extension of an

independent defect field

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 134

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect

  • fields. For instance, is every algebraic extension of an

independent defect field again an independent defect field?

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 135

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect

  • fields. For instance, is every algebraic extension of an

independent defect field again an independent defect field? Questions of this type are very hard to study when the valued field under consideration

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 136

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect

  • fields. For instance, is every algebraic extension of an

independent defect field again an independent defect field? Questions of this type are very hard to study when the valued field under consideration does not have a p-divisible value group

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 137

Independent defect fields

We call a valued field an independent defect field if it does not admit any dependent defect extensions of degree p. Hence by

  • ur earlier theorem, every gdr field is an independent defect

field. There are many open problems about independent defect

  • fields. For instance, is every algebraic extension of an

independent defect field again an independent defect field? Questions of this type are very hard to study when the valued field under consideration does not have a p-divisible value group or does not have a perfect residue field.

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 138

References

Gabber, O. – Ramero, L.: Almost ring theory, Lecture Notes in Mathematics 1800, Springer-Verlag, Berlin, 2003 Kuhlmann, F.-V.: A classification of Artin-Schreier defect extensions and a characterization of defectless fields, Illinois J.

  • Math. 54 (2010), 397–448.

Kuhlmann, F.-V.: The algebra and model theory of tame valued fields, J. reine angew. Math. 719 (2016), 1–43 Kuhlmann, F.-V. – Pal, K.: The model theory of separably tame fields, J. Alg. 447 (2016), 74–108 Temkin, M. : Inseparable local uniformization, J. Algebra 373 (2013), 65–119

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields

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SLIDE 139

Preprints and further information

The Valuation Theory Home Page http://math.usask.ca/fvk/Valth.html

Franz-Viktor Kuhlmann (joint work with Anna Blaszczok) Deeply ramified fields